1.6 元素訊號(elementary...
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1.6 元素訊號(Elementary signals)指數訊號(exponential signals)弦波訊號(sinusoidal signals)弦波訊號與複數指數訊號
指數衰退之弦波訊號
步階(Step)函數(訊號)脈衝(impulse)函數(訊號)
– 微分
RAMP function
ee: P34
指數訊號(exponential signals)
(a) 自然界中常見之衰退或成長函數, 如:人口自然成長函數, 放射物質之衰退
(b)基本函數型態
衰退
成長
指數蛻變常數
之初值
,0,0
:0:
,)(
<>
==
aaa
tBBetx at
ee: P34
圖例: 指數訊號(p34, F1.28)
(a) (b)
0,)( / >= − ττtBetx
0,)( / >= ττtBetx
ee: P34
計算: 一階RC電路
0)()(
0)()(
=+
=+
tvtvdtdRC
tvtRi
)(一階常微分方程式
)0(,)0()/(1,01
0
0)()(
)(
0 vBBevRCaRCa
BeRCaBe
tvtvdtdRC
Betv
a
atat
at
=∴=→
−=∴=+→=+
=+
=假設
)()0()( RCtevtv −=⇒
ee: P35
離散指數訊號(exponential signals)
(a) (b) (c) 類比於連續指數
10,][ <<= rBrnx n
1,][ >= rBrnx n
αer =
ee: P35
複數指數訊號
若B, r, a為複數(complex)則可定義出複數指數訊號
常見之複數指數訊號
n
at
BrnxBetx
=
=
][)(
( )njnj
tj
eBBenx
BetxΩΩ ==
=
][
)( ϖ
Fourier transform之基本基底訊號連續離散
ee: P36
弦波訊號(sinusoidal signals) ee: P36
)35.1()cos()( φϖ += tAtx
啟始相位
角頻率
振幅
重要參數
:2
::
φπϖ
ϖf
A
=
ϖππϖ
2,2
)()(
=→
∈==+
T
nnTtxTtx
Z
計算: 二階LC電路ee: P36
∫
∫
=+
=+
0)(1)(
0)(1)(
dttiC
tidtdL
dttiC
tv
0)()(2
2
=+ tvtvdtdLC
代入
二階常微分方程式
)cos()0()(0)0(),0(
)sin()cos(,,1)/(1,01
0)(
0
000
02
2
tvtviv
tjAtAaLC
jRLaRCa
BeBeLCaBetv
ir
atat
at
ϖ
ϖϖϖ
ϖ
=⇒=
+=→
=−=∴=+→
=+
=
代入初值
解型態為複數
假設振盪電路之自然頻率
離散弦波訊號 ee: P37,38
)cos(][ φ+Ω= nAnx
( )
整數∈=Ω
=Ω⇒++Ω=+
NmcycleradNmor
radiansmNNnANnx
,,2,2
)(cos][
ππ
φ
基本三角恆等式
( )( ))2cos(1)(sin
)2cos(1)(cos
212212
θθ
θθ
−=
+=
)cos()sin(2)2sin()(sin211)(cos2)2cos( 22
θθθθθθ
=−=−=
)sin()sin()cos()cos()cos()sin()cos()cos()sin()sin(φθφθφθφθφθφθ
=±±=±
[ ][ ][ ])sin()sin()cos()sin(
)cos()cos()cos()cos()cos()cos()sin()sin(
2121
21
φθφθφθφθφθφθφθφθφθ
++−=
++−=
+−−=
ee: P763,
A-1
Ex 1.7( ) [ ]
)cos(][][][(b)period) al(fundament(a)
)5cos(3,5sin][
21
21
φ
ππ
+Ω=+=
==
nAnxnxny
nnxnnx
求
求主週期
225,2
= →Ω
=
=Ω=Ω
NmN
mN
最小整數πππ
( )[ ])sin()sin()cos()cos()cos(][
)5cos(35sin][][][ 21
φφφππ
nnAnAnynnnxnxny
Ω−Ω=+Ω=+=+=
3)cos(
,1)sin(,5
=→
−=→=Ω→
φ
φπ
A
A( ) 2sin11)sin(
3,31)tan(
)cos()sin(
3 =−=→−=
−=∴−
==→
−πφ
πφφφφ
AAAA
ee: P38~39
Ex: 1.7
弦波訊號與複數指數訊號
)sin()cos( '
θθθ jeidentitysEuler
j +=
( )
tj
tj
tjtjjtj
jtj
BetABetA
AeeAeBeAeBBe
ϖ
ϖ
φϖϖφϖ
φϖ
φϖ
φϖ
Im)sin(Re)cos(
,
=+ →
=+ →
==
=+
取虛部
取實部
其中
複數指數訊號
離散弦波訊號與複數指數訊號ee: P41
Fig 1.34
( )
nj
nj
njnjjnj
jnj
BenABenA
AeeAeBeAeBBe
Ω
Ω
+ΩΩΩ
Ω
=+Ω →
=+Ω →
==
=
Im)sin(Re)cos(
,
φ
φ
φφ
φ
取虛部
取實部
其中
複數指數訊號
N=8
指數衰退之弦波訊號ee: P41
Fig 1.35
)sin()( φϖ += − tAetx at
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-40
-30
-20
-10
0
10
20
30
40
50
60
)()()()sin()(
)(
21
2
1
txtxtxttx
Aetx at
=+=
= −
φϖ
with A = 60 and α = 6.
計算: RLC電路ee: P42
Fig 1.36
∫ ∞−
tdv
Lττ )(1
0)(1)(1)(
:
=++ ∫ ∞−
tdv
Ltv
Rtv
dtdC
KCL
ττ
0)(1)(1)(2
2
=++ tvL
tvdtd
Rtv
dtdC
AAevCRLCRC
CRLCj
RCRLCLCRLLj
RjLjRLCAetv tj
==
−=−
=∴
−±−
=−±−
=+→
=++++
= +
0
22
22
22
2
)(
)0(4
11,2
14
112
12
4
0)()()(
初值
ωσ
ωσ
ϖσωσ
ϖσ假設解
節點總流出電流=0
)4( CLRassume >
單位步階訊號, unit-step signal
<≥
=00
,0,1
)(tt
tu
<≥
=00
,0,1
][nn
nu
ee: P43
Fig 1.38
0 1 2 3 4-1-2-3
以單位步階訊號表示下列訊號ee: P44
Fig 1.39
))5.0(()5.0()()5.0()(
2
1
−−=+=−=
tututxtutx
)5.0()5.0()()()( 12 −−+=−=∴ tututxtxtx
以步階訊號描述開關動作ee: P45
Fig 1.40
)()1()(
,0)(,0)0(
)/(1
,)(
)(0
0
0
tueVtv
vBvVBvVvv
RCavBevtv
RCt
ppp
pat
P
−−=
−=→==+→=∞=
−=
+=
為特解
描述開關動作後之輸出
離散脈衝(Discrete-time form of impulse.) ee: P46
Fig 1.41
][nδ
≠=
=00
,0,1
][nn
nδ
脈衝(Dirac delta) ee: P46
Fig 1.42
∫∞
∞−=
≠=
1)(
0 ,0)(
dtt
andtfort
δ
δ
)()( lim0
txt ∆→∆
=δ
Dirac delta & unit step
∫ ∞−=
=
tdtu
tudtdt
ττδ
δ
)()(
)()(
∫∞
∞−=
≠=
1)(
0 ,0)(
dtt
andtfort
δ
δ
<≥
=00
,0,1
)(tt
tu
Ex 1.10 ee: P47
Fig 1.43
)()()()(
)()(
0
0
tCVtituVtv
dttdvCti
δ=∴=
=
Dirac delta之時間變數運算ee: P48
∫∞
∞−−=− )()()(
:&
00 ttxtttx
shifting
δ
積分
0),(1)(
:
>= ata
at
scaling
δδ
Dirac delta 可視 為 一 shiftoperator
圖解脈衝訊號之scaling 運算ee: P48
Fig. 1.44
)(1)( ta
at δδ =
)(1)()( limlim00
txa
atxat ∆→∆
∆→∆
==δ
Parallel LRC ee: P49
Fig. 1.45
+= 0at )( voltage theof value thefind )( ttva
CIVCvtI
tvR
tvdtdCdttv
LtIa
ttdt
/,)0()(
)(1)()(1)( )(
)()(
00
0
00
0
=∴= →
++=
=
+
∞−
∞−
∫
∫+
+
δ
δ
δ
兩邊積分至
P1.23 series LCR ee: P49
Fig. 1.45
+
+
≥
=
0for ofequation aldifferenti-integro the write)(0at )(current theof value thefind )(
ti(t)bttia
LVILitV
tRidiC
tidtdLtVa
ttdt
/,)0()(
)()(1)()( )(
)()(
00
0
00
0
=∴= →
++=
=
+
∞−
∞−
∫
∫+
+
δ
ττδ
δ
兩邊積分至
0)()(1)(
)()(1)()( )(
0
0
=++→
++=
∫
∫
∞−
≥
∞−
+
tRidiC
tidtdL
tRidiC
tidtdLtVb
tt
t
ττ
ττδ
Derivatives of the Impulse( ) ( )[ ]22
1)()( lim0
)1( ∆−−∆+∆
==→∆
ttdt
tdt δδδδ
0
)()()(
0)(
0)1(
)1(
tt
tfdtddttttf
dtt
doublet
=
∞
∞−
∞
∞−
=−
=
∫
∫δ
δ
性質
( ) ( )∆
∆−−∆+=
=
→∆
22
)()(
)1()1(
0
2
2)1(
lim tt
tdtdt
dtd
doublet
δδ
δδ
微分
( ) ( ) 0221110
lim =
∫ ∆−−∫ ∆+∆
==
∞∞−
=
∞∞−
→∆
dttdtt δδ
( ) ( )[ ]( ) ( )[ ]
0
)(221
221)(
000
000
lim
lim
tt
tfdtdttfttf
dttttttf
=→∆
∞
∞−→∆
=∆−−−∆+−∆
=
∆−−−∆+−∆
= ∫ δδ
ee: P50
P1.24( ) ( )
∆∆−−∆+
==→∆
22)()()1()1(
0
)1()2( lim tttdtdt δδδδ
( ) ( )
0
)()()(
2)(2)(
)()()(
2020
0
0)1(
0)1(
0
0)2(
lim
lim
tt
tdd
tdd
tfdtd
dtdtftf
dttttfdttttf
dttttfa
=
∆−=∆+=
→∆
∞
∞−
∞
∞−
→∆
∞
∞−
=
∆
−−−=
∆
∆−−−∆+−=
−
∫∫∫
ττττ ττ
δδ
δ
性質)()2( tδ
ee: P51
( ) ( )
0
)(...
2)(2)(
)()()(
0)1(
0)1(
0
0)(
lim
ttn
n
nn
n
tfdtd
dttttfdttttf
dttttfb
=
∞
∞−
−∞
∞−
−
→∆
∞
∞−
==
∆
∆−−−∆+−=
−
∫∫∫
δδ
δ
Ramp Signal ee: P51,52
Fig. 1.46,47
)(00
,0,
)( ttuttt
tr =
<≥
=
][00
,0,
][ nnunnn
nr =
<≥
=
1 2 3 4-1-2
Ex 1.11 ee: P52
Fig 1.48
)()( 0 tuIti =
)(
)(1
)(1)(
0
0
trCI
duIC
diC
tv
t
t
=
=
=
∫
∫
∞−
∞−
ττ
ττ